Dynamical Systems in Neuroscience:

Bursting 3951K + activation gate, n0.80.60.40.20-80 -60 -40 -20 0membrane potential, V (mV)Figure 9.55: Bursting in two-dimensional I Na,p +I K -model with parameters as inFig. 6.16 and I = 43.in the point-point hysteresis loops in Ex. 19. This assumption is violated when the neuronfires a burst of spikes. Thus, the theory is helpless in studying fast-slow point-cyclebursters. An exception is Pontryagin’s problem, which is related to “fold cycle/foldcycle” bursting; see Ex. 21 below and Sect. 7 in Mishchenko et al. (1994). Pontryaginand Rodygin (1960) pioneered the method of averaging of the fast subsystem, whichwas used in the context of bursters by Rinzel and Lee (1986), Pernarowski et al. (1992),Smolen et al. (1993), Baer et al. (1995). Shilnikov et al. (2005) introduced an averagenullcline of the slow subsystem, and showed how the averaging method can be used tostudy co-existence of spiking and bursting states in a model neuron, and bifurcationsin bursters in general. Some of the transitions “resting ↔ bursting ↔ tonic spiking”were also considered by Ermentrout and Kopell (1986a), Terman (1991), Destexhe andGaspard (1993), Shilnikov and Cymbalyuk (2004, 2005), and Medvedev (2005).The averaging method, as many other classical methods of analysis of dynamicalsystems, breaks down when the fast subsystem slowly passes a bifurcation point. Thedevelopment of early dynamical system theory was largely motivated by studies ofperiodic oscillators. It is reasonable to expect that the next major developments ofthis theory will be coming from studies of bursters.Exercises1. (Planar burster) Invent a planar system of ODEs having a hedgehog limit cycleattractor, as in Fig. 9.54, and capable of exhibiting periodic bursting activity.2. (Noise-induced bursting) Explain why the I Na,p +I K -model with the phase portraitas in Fig. 9.55 bursts even though it has only two dimensions.3. (Noise-induced bursting) Explore numerically the I Na,p +I K -model with phaseportrait as in Fig. 6.7, top, and make it burst as in Fig. 9.56 without addingany new current or gating variable.